Starlike cases of the generalized goodman conjecture

Abstract

We consider functions f that are meromorphic and univalent in the unit disc $\mathbb{D}$ with a simple pole at the point p ∈ (0, 1) and normalized by f(0) = f′(0) − 1 = 0. A function g is called subordinated under such a function f, if there exists a function ω holomorphic in $\mathbb{D}$, ω($\mathbb{D}$) ⊂ , such that g(z) = f(zω(z)), z ∈ $\mathbb{D}$, and we use the abbreviation g ≺ f to indicate this relationship between two functions. We conjectured that for g ≺ f, the inequalities

$|a_n (g)| \leqslant \frac{1} {{p^{n - 1} }}\sum\limits_{k = 0}^{n - 1} {p^{2k} } ,n \in \mathbb{N}, $

are valid. Here f is as above and the expansion

$g(z)\sum\limits_{n = 1}^\infty {a_n (g)z^n } $

is valid in some neighbourhod of the origin. In the present article, we prove that this is true for two classes of functions f for which \f($\mathbb{D}$) is starlike.